There's this clunking/popping noise coming from the engine bay. It only happens once until you come to a stop. I've jacked the car up to look for any obvious loose bolts (including engine mount), and I don't see anything.

Yes, to clarify the noise only happens under moderate acceleration. Once the noise occurs it generally will not happen again until the car comes to a stop. The noise only occurs in first or second gear and the car is a manual. In the first part of the video the noise occurs while dropping the clutch in 1st (This is the only way I could get the noise to occur with the car jacked up). In the second part the noise occurs in 2nd with the clutch fully engaged under moderate acceleration.

The only inspection I have done is visual so I have not ruled that out. The thing that leads me to believe a motor mount is not at fault is that once the noise occurs I can't cause it to occur a second time unless I come to a stop (or at the very least change gears, as once it did happen in 1st then in 2nd without stopping). Am I wrong with this assumption?

Well with a bad mount....If it was loose enough to be causing a clunk you would think that you could get it to make the noise more than once without stopping.....So you may have something else going on but I would still want to be certain the mounts were good...

The best way to check them is to power brake the engine in forward & then reverse and look at the mount. Alot of times when they "split", the 2 broken pieces sit on top of each other so when you look at it, if you don't look real close the mount will look good. When you power brake the engine, you can see the 2 pieces seperate and it's very obvious.... Also if you get excessive engine movement in the compartment, is another sign of a weak or broken mount.....

Manauls are a little harder to check.....You have to have one foot on the brake, while popping the clutch slightly & giving it gas at the same time....... extreme caution needs to be used.....

If the mount is broken all the way.....a close inspection of rubber piece with a good light will show the "line" of the 2 broken pieces sitting on top of each other....